The last month has seen a plethora of discussions about the necessity for teaching math beyond what most jobs consider necessary. Much of it started from Andrew Hacker's now infamous article on whether math is necessary, to which a bunch of us replied with equal fervor (Dan Willingham's and Sherman Dorn's pieces are great rejoinders). What we all seem to agree on is that, indeed, the way we teach math matters. Lots. Having a positive environment for kids where they feel like they can actually do math without feeling like they're complete failures matters a lot.

Often, that starts with us as teachers.

Developing an environment where students can experiment and gain entry into the language of math starts with having a person who can facilitate what Stephen Krashen termed a low affective filter environment. While his study was applied to English Language Learners, his hypotheses should apply to all subject areas, math highest among them.

In my classroom, I have five principles for assuring that all students can enter into the math, and also for creating the conditions for math success.

1) Allow More Mistakes

I would suggest this to just about every teacher, but specifically math teachers, especially those of us who use the word "wrong" a lot. We should strike a balance between using direct instruction and exploration, leaning more on the exploration piece. Once we allow more mistakes, we let students into the process that our earliest mathematicians used in developing the axioms we believe today. Also, by admitting that we all make mistakes, it sends a clear signal to kids that they can be mathematicians, too. Surely, I'm not suggesting that we let the mistakes be. Yet, when I make a mistake on the board (intentionally or otherwise), I hope my students catch onto that, thus putting them in the position of expert. Speaking of which . . .

2) Support Their Struggle

At first, most of us get nervous when children struggle with mathematics, as if they need to get the math as soon as they receive the instruction. Even if it might look simple to us, the students may still be grappling with the skills, the concepts or both. That's OK. When students struggle with the material, they learn how to work problems out on their own as self-motivated workers. Of course, that also means the teacher needs to encourage them as often as possible to do so. If students think their efforts have no merit, then they often won't own it. In an environment where teachers support students' working through a problem, a teacher can tell when a student has quit. You have an option before you intervene . . .

3) Let the Kids Teach, Too

During the class period, I prefer the students speak more than I do. If I'm talking too much, that means I'm using too much of my speaking quota. In other words, they'll tune me out if I'm talking too much. Once I let the students speak (meaning, not just one student, but many), they take even more ownership of the math taught to them. This especially proves true during the class work time as well. Having them explain to each other (with the proper guidance) really empowers them to own the material and develop their own process for checking answers. Plus, I’m not exceeding my speaking quota. However . . .

4) Answer a Question with More Questions

If, in fact, a student asks us a question, we ought to validate their question by giving them another question. That way, we ensure that the onus for the "answer" falls on them. The type of questions we ask and the way we frame questions matters here, too. Questions that generate a "yes" or "no" answer simply won't do. Instead, we can leave them with a question that they can answer. I do emphasize the word "leave" because it's always good for you to walk away without explicitly telling them they were right. By the time you leave, they should already know this.

5) Personalize the Questions

Inserting children's names into the problems (appropriately) engages students in the material. As you start the problem, speaking about the student in the third person immediately gets him or her engaged, and gives the other students a window into the problem. Knowing the person in the problem (even if the situation itself is hypothetical) gives the entire class a sense of ownership and belonging within the math. Obviously, teachers should spread the wealth in naming people, because it may look like we're playing favorites or just focusing on the "struggling" children. All children need access to the math.

This advice obviously takes time and a teacher's willingness to fail. This also might push some of my colleagues out of a comfort zone. Yet we as teachers have to set a precedent for the success of lifelong learners, not just until a standardized test comes. One of the ways we set our students up for this type of success is by providing conditions for questioning, experimentation and ownership to happen. Those of us who only want to "skill and drill" students perpetuate attitudes that Hacker alluded to in his piece.

Reading and English teachers have known for a long time that correcting every error while a child learns to read and write only causes frustration and anxiety. More math teachers need to recognize that the same principle applies in math. I agree with you comments regarding allowing errors as a part of the learning process. We need to focus on the important concepts of the lesson and not fret so much that there may be unrelated errors. Perfection comes with practice.

Jose, great article. For the month of October, I am giving my book Teen Juggernaut away FREE to anyone that emails me at robleegarcia@yahoo.com and asks for it. Its in conjunction with Anti Bullying Month but the book has a significant math section as well that breaks down WHY students need to learn math, and how it can turn into a high paying career. I talked about the highest paying jobs right out of college and the relationship between math and getting into these fields. There is also a chapter on better studying habits and developing strategies for tackling math. If you or any Edutopia readers are interested in my book, email me and Ill send it out immediately. Thanks,
Rob.

This all sounds very exciting and promising to me. Now I want to know, what does "allowing more mistakes" look like? Can you describe a scenario or tell a story in which mistakes are allowed so that I can see myself doing it?

I enjoyed reading your article. I thoroughly agree with your statement, "Having a positive environment for kids where they feel like they can actually do math without feeling like they're complete failures matters a lot, (Vilson, 2012).
You list five very good reasons that allow your students to enter into a safe math classroom. Allowing students to make more mistakes in the classroom was my favorite. I thoroughly agree that we should allow our student to make more mistakes in the classroom. It is the math teacher that needs to instill in their student a different type of critical thinking when solving problems that allows for mistakes and creativity. The students need to feel that if they make a mistake that it is ok for them to correct their mistake and maybe come up with a new method for solving a math problem. As math teachers we know that there is more than one way to solve a problem. I am going to try this method out in my math classes.

Your reference to Stephen Krashen is what first caught my attention. Yes! Lowering the affective filter is not an idea that I limit to my instruction of English Language Learners. Math is, in fact, a language all its own and it makes sense that we treat it as such. Your five principles were extremely helpful. All principles really resonated with me but the one that I will definitely take into my classroom starting tomorrow is to "Let Kids Teach, Too." Our students have the capability to offer so much to their peers if we only give them a chance. I have found that once a lesson is done and the independent practice has started, the early finishers, once I have checked their work, could be the ones I call upon to peer tutor their classmates. This not only gives them a chance to deepen their own learning by having to help/explain/tutor a peer but it also gives struggling students the opportunity to work in a situation where that affective filter is lowered by dint of working with someone their own age, who has the same or almost the same experience with the subject matter that they do. It's like they're getting a second chance at the assignment...and I am always there to help as well. It seems they find it easier to go to a peer and ask for help than to ask the teacher. I am okay with that, as long as I can monitor what is happening and intervene when I need to. Thanks so much for your ideas!

Thank you for your great ideas, I plan on incorporating them in my teaching practices. I agree with you about creating conditions for success to promote student engagement. I find that when students feel confident about a given task they remain engage. One way that I try to build confidence is by providing tasks that allow students to have access from multiple entry points. Creating tasks that builds from prior knowledge also help students feel confident about what they are learning.

Your five principles for your classroom are great reminders for math teachers! It is amazing to me how many of my students say things like, "I hate math," and "I'm just not good at math, but neither is my mother." I, too, try to create a positive environment where mistakes are just the fact of life when working math problems. I am going to take your advice and try to use the word "wrong" less. I also especially liked your idea to answer the students' questions with another question. This sounds like a good way to continue their own thinking processes.